Recursive Contracts and Endogenously Contracts and Endogenously Incomplete Markets ... and martingale methods in continuous ... the methods of recursive contracts can

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<ul><li><p>Recursive Contracts</p><p>and Endogenously Incomplete Markets</p><p>Mikhail Golosov, Aleh Tsyvinski and Nicolas Werquin</p><p>January 2016</p><p>Abstract</p><p>In this chapter we study dynamic incentive models in which risk sharing is endoge-</p><p>nously limited by the presence of informational or enforcement frictions. We comprehen-</p><p>sively overview one of the most important tools for the analysis such problems the theory</p><p>of recursive constracts. Recursive formulations allow to reduce often complex models to a</p><p>sequence of essentially static problems that are easier to analyze both analytically and com-</p><p>putationally. We first provide a self-contained treatment of the basic theory: the Revelation</p><p>Principle, formulating and simplifying the incentive constraints, using promised utilities as</p><p>state variables, and analyzing models with persistent shocks using the first-order approach.</p><p>We then discuss more advanced topics: duality theory and Lagrange multiplier techniques,</p><p>models with lack of commitment, and martingale methods in continuous time. Finally, we</p><p>show how a variety of applications in public economics, corporate finance, development and</p><p>international economics featuring incompete risk-sharing can be analyzed using the tools of</p><p>the theory of recursive contracts.</p><p>Golosov: Princeton University; Tsyvinski: Yale University; Werquin: Toulouse School of Economics. Golosovand Tsyvinski thank the NSF for financial support.</p></li><li><p>1 Introduction</p><p>Dynamic incentive problems are ubiquitous in macroeconomics. The design of social insurance</p><p>programs by governments, long-run relationships between banks and enterpreneurs, informal in-</p><p>surance contracts against idiosyncratic shocks provided in village economies, sovereign borrowing</p><p>and lending between countries can all be understood using the theory of dynamic incentives.</p><p>These models have been widely used in macroeconomics, public economics, international macro,</p><p>finance, developement, or political economy, both for explaining existing patterns in the data and</p><p>for normative policy analysis. The unifying feature of these models is that, at their essence, they</p><p>study endogenously incomplete markets, i.e., environments in which risk sharing is constrained</p><p>by (informational or enforcement) frictions and insurance arrangements arise endogenously.</p><p>One of the most important tools used for studying dynamic incentive problems is the the-</p><p>ory of recursive constracts. Recursive formulations allow one to reduce often complex models</p><p>to a sequence of essentially static problems that are easier to analyze both analytically and</p><p>computationally. This substantially simplifies the analysis and the characterization of the op-</p><p>timal insurance arrangements in rich and realistic environments. The goal of this chapter is</p><p>to provide an overview of the theory of recursive contracts and give a number of examples of</p><p>application. The analysis in the theoretical part is self-contained; whenever a textbook approach</p><p>is not directly applicable (e.g., when the assumptions needed to apply the recursive techniques</p><p>in Stokey et al. (1989) are not met), we provide the necessary mathematical background. We</p><p>also discuss the strengths and weaknesses of several alternative approaches to solving dynamic</p><p>incentive problems that emerged in the literature. In the last part of the chapter we show how</p><p>the methods of recursive contracts can be used in a variety of applications.</p><p>Our paper is organized as follows. Section 2 considers a prototypical dynamic incentive</p><p>problem insurance against privately observable idiosyncratic taste shocks under perfect com-</p><p>mitment by the principal. The goal of this section is to provide an example of a self-contained,</p><p>rigorous and relatively general treatment of a dynamic incentive problem. We also use this</p><p>economy in subsequent sections to illustrate other approaches to the analysis of dynamic incen-</p><p>tive problems. In Section 2 we highlight the three main steps in the analysis: first, applying</p><p>the Revelation Principle to set up a mechanism design problem with incentive constraints; sec-</p><p>ond, simplifying this problem by focusing on one-shot incentive constraints; and third, writing</p><p>this problem recursively using promised utilities as state variables. We then show how this</p><p>recursive formulation can be used to characterize the properties of the optimal insurance ar-</p><p>rangements in our economy. We derive general features of the optimal insurance contract and</p><p>characterize the long-run behavior of the economy in Section 2.4. We show how to overcome</p><p>the technical difficulties that arise when the idiosyncratic shocks are persistent in Section 2.5,</p><p>and conclude by showing how the same techniques can be applied to other dynamic incentive</p><p>problems, such as moral hazard.</p><p>Section 3 considers more advanced topics. We focus on three topics: using Lagrange mul-</p><p>1</p></li><li><p>tiplier tools in recursive formulations, studying dynamic insurance problems in economies in</p><p>which the principal has imperfect commitment, and applying martingale techniques to study</p><p>recursive contracts in continuous time. Section 3.1 discusses the Lagrangian techniques. Us-</p><p>ing Lagrangians together with the recursive methods of Section 2 greatly expands the class of</p><p>problems that can be characterized. We first provide an overview of the theory of constrained</p><p>optimization using Lagrange multipliers, with a particular focus on showing how to use them in</p><p>the infinite dimensional settings that frequently arise in macroeconomic applications. We then</p><p>show how to apply these theoretical techniques to incentive problems to obtain several alterna-</p><p>tive recursive formulations having some advantages relative to those discussed in Section 2. A</p><p>number of results in this section are new to the dynamic contracts literature. In Section 3.2 we</p><p>show how to analyze dynamic insurance problems in settings where the principal cannot commit</p><p>to the contracts. The arguments used to prove simple versions of the Revelation Principle under</p><p>commitment fail in such an environment; we discuss several ways to generalize it and write a</p><p>recursive formulation of the mechanism design problem. Our characterization of such problems</p><p>relies heavily on our analysis of Section 3.1. Finally, in Section 3.3, we show how to analyze</p><p>a dynamic contracting problem in continuous time using martingale methods and the dynamic</p><p>programming principle. To keep the analysis self-contained, we start by stating the results of</p><p>stochastic calculus that we use. Continuous-time methods often simplify the characterization of</p><p>optimal contracts, allowing for analytical comparative statics and easier numerical analysis of</p><p>the solution.</p><p>Section 4 gives a number of applications of the recursive techniques discussed in Sections 2</p><p>and 3 to various environments. We show that these diverse applications share three key features:</p><p>(i) insurance is endogenously limited by the presence of a friction; (ii) the problem is dynamic;</p><p>(iii) the recursive contract techniques that we develop in the theoretical sections allow us to</p><p>derive deep characterizations of these problems. We explain how theoretical constructs such as</p><p>the incentive constraints and promised utilities can be mapped into concrete economic concepts,</p><p>and how the predictions of dynamic incentive models can be tested empirically and used for</p><p>policy analysis. In Section 4.1, we apply the techniques and results of Section 2 to public</p><p>finance where the endogenous market incompleteness and the limited social insurance arise</p><p>due to the unobservability of the shocks that agents receive. We derive several central results</p><p>characterizing the optimal social insurance mechanisms and show how to implement the optimal</p><p>allocations with a tax and transfer system that arises endogenously, without restricting the</p><p>system exogenously to a specific functional form. In Section 4.2 we show how recursive techniques</p><p>can be applied to study the effect of informational frictions on firm dynamics and optimal capital</p><p>structure. Section 4.3 presents applications of these techniques to study insurance in village</p><p>economies in developing countries where contracts are limited by enforcement and informational</p><p>frictions. Section 4.4 discusses applications to international borrowing and lending.</p><p>2</p></li><li><p>2 A simple model of dynamic insurance</p><p>In this section we study a prototypical model of dynamic insurance against privately-observed</p><p>idiosyncratic shocks. Our goal is to explain the key steps in the analysis and the main economic</p><p>insights in the simplest setting. The mathematical techniques that we use as well as the economic</p><p>insights that we obtain extend to many richer and more realistic environments. We discuss</p><p>examples of such environments in following sections.</p><p>2.1 Environment</p><p>We consider a discrete-time economy that lasts T periods, where T may be finite or infinite.</p><p>The economy is populated by a continuum of ex-ante identical agents whose preferences over</p><p>period-t consumption ct 0 are given by tU (ct), where t R+ is an idiosyncratic tasteshock that the individual receives in period t, and U is a utility function.</p><p>Assumption 1. The utility function U : R+ R is an increasing, strictly concave, differen-tiable function that satisfies the Inada conditions limc0 U</p><p> (c) = and limc U (c) = 0.</p><p>All agents have the same discount factor (0, 1). In each period the economy receives eunits of endowment which can be freely transfered between periods at rate .</p><p>The idiosyncratic taste shocks are stochastic. We use the notations t = (1, . . . , t) t todenote a history of realizations of shocks up to period t and t</p><p>(t)</p><p>to denote the probability of</p><p>realization of history t. We assume that the law of large numbers holds so that t(t)</p><p>is also</p><p>the measure of individuals who experienced history t.1 An individual privately learns his taste</p><p>shock t at the beginning of period t. Thus, at the beginning of period t an agent knows his</p><p>history t of current and past shocks, but not his future shocks. This implies that his choices in</p><p>period t, and more generally all the period-t random variables xt that we encounter, can only</p><p>be a function of this history.</p><p>Some parts of our analysis use results from probability theory and require us to be more</p><p>formal about the probability spaces that we use. A standard way to formalize these stochastic</p><p>processes is as follows.2 Let T be the space of all histories T and let T be a probability</p><p>measure over the Borel subsets B(T)</p><p>of T . Thus,(T ,B</p><p>(T), T</p><p>)forms a probability</p><p>space. Any period-t random variable is required be measurable with respect to B(t), that</p><p>is, for any Borel subset M of R, x1t (M) = B Tt, where B is a Borel subset of t. Thisformalizes the intuition that the realization of shocks in future periods is not known as of period</p><p>t.</p><p>Until Section 2.5.1 we make the following assumptions about the idiosyncratic taste shocks:</p><p>Assumption 2. The set R+ of taste shocks is discrete and finite with cardinality ||.Agents shocks evolve according to a first-order Markov process, that is, the probability of drawing</p><p>1The assumption that the law of large numbers holds can be justified formally (see Uhlig (1996), Sun (2006)).2See Stokey et al. (1989, Chapter 7) for a review of the measure-theoretic apparatus.</p><p>3</p></li><li><p>type t in period t depends only on the period-(t 1) type:</p><p>t(tt1 ) = (t |t1 ) , t1 t1, t ,</p><p>where t1 is the last component of t1.</p><p>We use the notation t(t |s</p><p>)for t &gt; s to denote the probability of realization of history t</p><p>up to period t conditional on the realization of history s up to period s, with a convention that</p><p>t(t |s</p><p>)= 0 if the first s elements of t are not s (history t in period t cannot occur if s</p><p>was not realized up to period s). We use ts to denote (s, . . . , t). Finally we index the elements</p><p>of by the subscript (j) for j {1, . . . , ||}, and assume that (1) &lt; (2) &lt; . . . &lt; (||).We consider the problem of a social planner who chooses consumption allocations3 ct : </p><p>t R+ to maximize agents ex-ante expected utility and has the ability to commit to such allocationsin period 0. At this stage we are agnostic about who this planner is. One can think of it as</p><p>a government that provides insurance to agents, or as some decentralized market arrangement.</p><p>We study the optimal insurance contract that such a planner can provide given the feasibility</p><p>constraint and informational constraints. We use the shortcut c to denote the consumption plan{ct(t)}</p><p>t1,tt .</p><p>The ex-ante, period-0 expected utility of all agents is denoted by U0 (c) and is given by</p><p>U0 (c) E0</p><p>[Tt=1</p><p>t1tU (ct)</p><p>]=</p><p>Tt=1</p><p>tt</p><p>t1t(t)tU</p><p>(ct(t)). (1)</p><p>Here E0 represents the (unconditional) expectation at time 0, before the first-period type 1is known. Under our assumption that resources can be freely transferred between periods, the</p><p>resource constraint is</p><p>E0</p><p>[Tt=1</p><p>t1ct</p><p>] 1 </p><p>T</p><p>1 e. (2)</p><p>Note that to write the left hand side of this feasibility constraint we again implicitly invoke the</p><p>law of large numbers.</p><p>When the realizations of the taste shocks are observable by the planner, this problem can</p><p>easily be solved explicitly. Let &gt; 0 be the Lagrange multiplier on the feasibility constraint.</p><p>The optimal allocation cfb in the case where shocks are observable (the first best allocation)</p><p>is a solution to</p><p>tU(cfbt(t))</p><p>= , t 1, t t. (3)</p><p>It is immediate to see that this equation implies that cfbt(t)</p><p>is independent of period t or the</p><p>past history of shocks t1, and only depends on the current realization of the shock t. That is,</p><p>the informationally-unconstrained optimal insurance in this economy gives agents with a higher</p><p>3Formally, ct is a random variable over the probability space(T ,B</p><p>(T), T</p><p>)that is measurable with respect</p><p>to B(t).</p><p>4</p></li><li><p>realization of the shock in any period (hence whose current marginal utility of consumption</p><p>is higher) more consumption than agents with a lower realization of a taste shock.</p><p>2.2 The Revelation Principle and incentive compatibility</p><p>We are interested in understanding the properties of the best insurance arrangements that a</p><p>planner can provide in the economy with private information. This insurance can be provided</p><p>by many different mechanisms: the agents may be required to live in autarky and consume their</p><p>endowment, or may be allowed to trade assets, or may be provided with more sophisticated</p><p>arrangements by the planner. A priori it is not obvious how to set up the problem of finding</p><p>the best mechanism to provide the highest utility to agents. This problem simplifies once we</p><p>apply the results of the mechanism design literature, in particular the Revelation Principle.</p><p>Textbook treatments of the Revelation Principle are widely available (see, e.g. Chapter 23 in</p><p>Mas-Colell et al. (1995)). Here we outline the main arguments behind the Revelation Principle</p><p>in our context. This overview is useful both to keep the analysis self-contained and to emphasize</p><p>subtleties that emerge in using the Revelation Principle once additional frictions, such as lack</p><p>of commitment by the planner, are introduced.</p><p>Hurwicz (1960, 1972) provided a general framework to study vari...</p></li></ul>

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